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Math Help - Circular Curves and Tangents Lines

  1. #1
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    Circular Curves and Tangents Lines

    Hi MHF
    I'm having some hard times with circular functions like

    Let A be a point on the curve C: X^2+Y^2-2X -4=0. If the tangent line to C at A passes through point P(4,3), then the length of AP is:


    It seems hard to even see the image in my mind, I'm confused!

    Thanks in advance!
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  2. #2
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    x + y - 2x - 4 = 0 is equivalent to (x - 1) + y = √5.
    Line passing through the point P(4, 3) is y - 3 = m(x - 4), i.e. y = m(x - 4) + 3.
    Plug in to the circle equation, and we get 0 = (1 + m)x + (6m - 2 - 8m)x + (16m - 24m + 5).
    According to discriminant, b = 4ac.
    (6m - 2 - 8m) = 4(1 + m)(16m - 24m + 5), i.e. m = (9 √65)/4.
    Plug in y = (9 √65)(x - 4)/4 + 3 to the circle equation to get A((11 minusorplus √65)/6, (5 √65)/6).
    The length of AP is √13.
    Last edited by johnny; May 20th 2011 at 06:30 PM.
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  3. #3
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    Hi johnny

    Thank you so much for your answer!

    I managed to solve the problem in a different way.
    But I really like to see other ways to see the solution.
    I learn a lot this way!

    Thank you!
    Really appreciate your help.
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